simplicial vs (cubical with connections) Ronnie Brown

simplicial vs (cubical with connections) Ronnie Brown

[F William Lawvere]

 		categories: Re: Simplicial versus (cubical with connections)‏

 	Ronnie Brown
ronnie.profbrown@btinternet.com




 			 
 		
 	One basic intuition is that 
 				cubes are products.
 	Yet almost none of the combinatorial toposes commonly called
 	“cubical sets” have that feature.  Is there some profound disadvantage
 	in allowing projection maps to have their universal property (yielding
 	diagonal maps, etc)? If so, I have never seen it spelled out.
	
 	An advantage to having finite products in a site is that the (iterated)
 	pathspace functor has a right adjoint, leading to a very natural construction of Eilenberg-Mac Lane spaces.

 	Dan Kan told me that the reason for his switch was that cubical groups do not have the extension property (i.e., the Kan property), as simplicial groups do. 
 	But later I realized that there is ambiguity about what “cubical” means.

 	Since these combinatorial categories are usually toposes, some light is shed on their particularity by determining what kind of structure they classify 
(in the established categorical sense, e.g.,the simplicial topos classifies  total orders with distinct endpoints, and a simple cubical example classifies
  strictly bipointed objects). Concretely, there are many different theories  of algebraic 	structure for which the unit interval is a model, and having  chosen one,
this structure should be preserved by geometric realization.

 	Bill Lawvere




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